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\begin{document}
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\begin{center}\textbf{Math 31 Homework 1}\\Due June 29, 2018\end{center}



\begin{enumerate}

\item Are the following binary operations? Why or why not?
\begin{enumerate}
\item $a*b=a\ln(b)$ on the set of positive real numbers ($\{x\in\R:x>0\}$).
\item $a*b=|a-b|$ on the set of non-negative integers ($\{n\in\Z:n\geq 0\}$).
\end{enumerate}

\item Apply the directions from Chapter 2, exercise B to the operation $x*y=x+y-xy$.

\item Do the following Cayley tables represent groups? Why or why not? If so, is the group represented abelian? Why or why not? (If a Cayley table has both an identity and inverses, then I have made it associative - you do not need to check this. Ordinarily, you may not assume that an operation is associative, but we do not know any good ways aside from brute force to check associativity in Cayley tables, and I don't want to make you do that!)

\begin{enumerate*}
\item 
\begin{tabular}{ c | c c c c }
&a&b&c&d\\
  \hline			
  a&a & b & c&d \\
  b&c & b & a&d \\
  c&b & a & d&c \\
	d&d & c & b&a
\end{tabular}

\hspace{5mm}

\item \begin{tabular}{ c | c c c c }
&a&b&c&d\\
  \hline			
  a&a & b & c&d \\
  b&b & a & d&c \\
  c&c & c & d&b \\
	d&d & b & a&c
\end{tabular}

\hspace{5mm}

\item \begin{tabular}{ c | c c c c }
&a&b&c&d\\
  \hline			
  a&c & a & d&b \\
  b&a & b & c&d \\
  c&d & c & b&a \\
	d&b & d & a&c
\end{tabular}
\end{enumerate*}

\item Write out the Cayley table(s) for all group(s) with two elements (call them $a$ and $b$).
\begin{enumerate}
\item For each that you found, prove that it is a group.
\item Why are the other possible Cayley tables not groups? (you do not need to prove that each individual table is not a group, but do explain why you did not include them in your list).
\item We discussed $\Z_n$ in class. Make a Cayley table for $\Z_2$. Is this Cayley table the same as one that you found above? Which one? Why are they the same?
\item What conclusion can you make about the number of distinct groups with two elements?
\end{enumerate}

\item We talked about $D_4$ in class. For $D_3$ (symmetries of an equilateral triangle), do the following:
\begin{enumerate}
\item Describe the group elements.
\item Write out the Cayley table.
\item Select two triples and show that they are associative (this is not enough to prove that the whole operation is associative).
\item Prove that $D_3$ is a group (you may assume associativity).
\item Based on this problem and what we did with $D_4$ in class, how many elements do you expect $D_n$, the group of symmetries on an $n$-gon, to have? What do you expect these elements to look like?
\end{enumerate}


\end{enumerate}
\end{document}